explain how the​ chi-square independence test and the​ chi-square goodness-of-fit test are similar. How are they​ different?

Question content area bottom
Part 1
For each​ characteristic, determine if it applies to the​ chi-square independence​ test, the​ chi-square goodness-of-fit​ test, or both.
Has d.f.​(r​1)(c​1):

Chi-square independence test
Chi-square goodness-of-fit test
Both
Part 2
Requires the data be obtained from a random​ sample:

Chi-square independence test
Chi-square goodness-of-fit test
Both
Part 3
Has d.​1:

Chi-square independence test
Chi-square goodness-of-fit test
Both
Part 4
Uses this formula to find the expected​ frequency: ​:

Chi-square independence test
Chi-square goodness-of-fit test
Both
Part 5
Uses the following formula to find the expected​ frequency:

Chi-square independence test
Chi-square goodness-of-fit test
Both

Part 6
Requires that each expected frequency is at least​ 5:

Chi-square independence test
Chi-square goodness-of-fit test
Both
Part 7
Used to test if two variables are​ independent:

Chi-square independence test
Chi-square goodness-of-fit test
Both
Part 8
Test if a frequency distribution fits an expected​ distribution:

Chi-square independence test
Chi-square goodness-of-fit test
Both

The chi-square independence test and the chi-square goodness-of-fit test are both statistical tests that use the chi-square distribution to determine if there is a significant association between variables or if a frequency distribution fits an expected distribution.

Here is the breakdown of how they are similar and different:

Similarities:
1. Both tests use the chi-square distribution to calculate the test statistic.
2. Both tests require the data to be obtained from a random sample.
3. Both tests have degrees of freedom, denoted as df.

Differences:
1. The chi-square independence test is used to test the independence between two categorical variables. It determines if there is a relationship between two variables by comparing the observed frequencies in a contingency table to the frequencies that would be expected if the variables were independent.
2. The chi-square goodness-of-fit test is used to test whether an observed frequency distribution fits an expected distribution. It compares the observed frequencies in a single categorical variable to the frequencies that would be expected based on a specified distribution or set of proportions.
3. The chi-square independence test uses the formula for calculating expected frequencies as E = (row total * column total) / grand total.
4. The chi-square goodness-of-fit test uses the formula for calculating expected frequencies as E = (sample size * specified proportion).

Now, let's go through the characteristics and determine if they apply to the chi-square independence test, the chi-square goodness-of-fit test, or both:

Part 1: Has d.f. (r1)(c1)
- This applies to the chi-square independence test.

Part 2: Requires the data be obtained from a random sample
- This applies to both the chi-square independence test and the chi-square goodness-of-fit test.

Part 3: Has d.f. k1
- This applies to both the chi-square independence test and the chi-square goodness-of-fit test.

Part 4: Uses the formula to find the expected frequency: E = (row total * column total) / grand total
- This applies to the chi-square independence test.

Part 5: Uses the following formula to find the expected frequency: E = (sample size * specified proportion)
- This applies to the chi-square goodness-of-fit test.

Part 6: Requires that each expected frequency is at least 5
- This applies to both the chi-square independence test and the chi-square goodness-of-fit test.

Part 7: Used to test if two variables are independent
- This applies to the chi-square independence test.

Part 8: Test if a frequency distribution fits an expected distribution
- This applies to the chi-square goodness-of-fit test.

I hope this explanation clarifies the similarities and differences between the chi-square independence test and the chi-square goodness-of-fit test, as well as the characteristics associated with each test.